Subgroups — #1336×C₂₅₁ #44545923

Order 32128.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#443×C₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#364×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#322×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#135×C₂₅₁	C₄	1	1	yes	characteristic
5	#131×C₂₅₁	—	2	1	no	—
6	M₃₂×C₂₅₁	—	2	1	no	—
7	#108×C₂₅₁	C₄	1	1	yes	characteristic
8	#104×C₂₅₁	—	2	1	no	—
9	C₄×C₈×C₂₅₁	—	2	1	no	—
10	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
11	#41×C₂₅₁	—	2	1	no	—
12	Q₁₆×C₂₅₁	C₈	1	1	yes	characteristic, complemented
13	SD₁₆×C₂₅₁	—	2	1	no	—
14	D₈×C₂₅₁	C₈	1	1	yes	characteristic, complemented
15	M₁₆×C₂₅₁	—	2	1	no	—
16	M₁₆×C₂₅₁	—	4	1	no	—
17	C₂×C₈×C₂₅₁	—	2	2	no	—
18	C₂×C₈×C₂₅₁	—	4	1	no	—
19	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
21	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
22	C₁₆×C₂₅₁	—	4	1	no	—
23	Q₈×C₂₅₁	—	2	1	no	—
24	D₄×C₂₅₁	—	2	1	no	—
25	D₄×C₂₅₁	—	4	1	no	—
26	C₂×C₄×C₂₅₁	—	2	1	no	—
27	C₂×C₄×C₂₅₁	—	4	1	no	—
28	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₅₁	—	4	2	no	—
30	C₈×C₂₅₁	—	4	1	no	—
31	C₈×C₂₅₁	C₂×C₈	1	1	yes	characteristic
32	C₈×C₂₅₁	M₁₆	1	1	yes	characteristic
33	C₈×C₂₅₁	D₈	1	1	yes	characteristic
34	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
35	C₂²×C₂₅₁	—	4	1	no	—
36	C₂²×C₂₅₁	#104	1	1	yes	characteristic
37	C₄×C₂₅₁	—	2	2	no	—
38	C₄×C₂₅₁	—	4	1	no	—
39	C₄×C₂₅₁	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₂₅₁	#102	1	1	yes	characteristic
41	C₂×C₂₅₁	—	2	1	no	—
42	C₂×C₂₅₁	—	8	1	no	—
43	C₂×C₂₅₁	D₄⋊C₈	1	1	yes	characteristic
44	C₂₅₁	#1336	1	1	yes	characteristic, direct product factor, Sylow
45	#1336	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	#443	C₂×C₂₅₁	1	1	yes	characteristic
47	#364	C₂×C₂₅₁	1	1	yes	characteristic, complemented
48	#322	C₂×C₂₅₁	1	1	yes	characteristic, complemented
49	#135	C₄×C₂₅₁	1	1	yes	characteristic
50	#131	—	2	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	C₄×C₂₅₁	1	1	yes	characteristic
53	#104	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	C₂²×C₂₅₁	1	1	yes	characteristic
56	#41	—	2	1	no	—
57	Q₁₆	C₈×C₂₅₁	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	C₈×C₂₅₁	1	1	yes	characteristic, complemented
60	M₁₆	—	2	1	no	—
61	M₁₆	—	4	1	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	C₂×C₄×C₂₅₁	1	1	yes	characteristic
65	C₂×C₈	D₄×C₂₅₁	1	1	yes	characteristic
66	C₄²	D₄×C₂₅₁	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
74	C₈	—	4	2	no	—
75	C₈	—	4	1	no	—
76	C₈	C₂×C₈×C₂₅₁	1	1	yes	characteristic
77	C₈	M₁₆×C₂₅₁	1	1	yes	characteristic
78	C₈	D₈×C₂₅₁	1	1	yes	characteristic
79	C₈	SD₁₆×C₂₅₁	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#104×C₂₅₁	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	C₂²⋊C₈×C₂₅₁	1	1	yes	characteristic
85	C₄	#102×C₂₅₁	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊C₈×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₅₁	#102
commutator	C₈	C₂×C₈×C₂₅₁
Frattini, cosocle	C₄×C₈	C₂²×C₂₅₁
socle	C₂×C₂₅₁	D₄⋊C₈
prime radical	#135×C₂₅₁	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₅₁ ◁ C₂×C₈×C₂₅₁ ◁ C₄×C₈×C₂₅₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #364×C₂₅₁	C₂	semidirect
3. #322×C₂₅₁	C₂	semidirect
12. Q₁₆×C₂₅₁	C₈	semidirect
14. D₈×C₂₅₁	C₈	semidirect
44. C₂₅₁	#1336	direct (N × H)
45. #1336	C₂₅₁	direct (N × H)
47. #364	C₂×C₂₅₁	semidirect
48. #322	C₂×C₂₅₁	semidirect
57. Q₁₆	C₈×C₂₅₁	semidirect
59. D₈	C₈×C₂₅₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#443×C₂₅₁	2
#364×C₂₅₁	2
#322×C₂₅₁	2
#1336	251

Subgroups — #1336×C251 #44545923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1336×C₂₅₁ #44545923